The diffusive dynamics of a particle in a medium with space-dependent friction coefficient is studied within the framework of the inertial Langevin equation. In this description, the ambiguous interpretation of the stochastic integral, known as the Itô-Stratonovich dilemma, is avoided since all interpretations converge to the same solution in the limit of small time steps. We use a newly develo...

We give a summary on the geometry of iterated Stratonovich integrals. For this exposition, we always have the connection to stochastic Taylor expansion in mind. In particular, we believe that “cubature on Wiener space” is best understood in the setting presented in this text. Besides cubature on Wiener space, we also give a second application regarding the heat kernel on nilpotent free Lie groups.

We study the complexity of approximating stochastic integrals with error ε for various classes of functions. For Ito integration, we show that the complexity is of order ε−1, even for classes of very smooth functions. The lower bound is obtained by showing that Ito integration is not easier than Lebesgue integration in the average case setting with the Wiener measure. The upper bound is obtaine...

In this paper we extend the recent work of C.A. Braumann [1] to the case of stochastic differential equation with random coefficients. Furthermore, the relationship of the Itô-Stratonovich stochastic calculus to studies of random population growth is also explained.

In this paper, we apply Legendre wavelet collocation method to obtain the approximate solution of nonlinear Stratonovich Volterra integral equations. The main advantage of this method is that Legendre wavelet has orthogonality property and therefore coefficients of expansion are easily calculated. By using this method, the solution of nonlinear Stratonovich Volterra integral equation reduces to...

We explain how Itô Stochastic Differential Equations on manifolds may be defined as 2-jets of curves. We use jets as a natural language to express geometric properties of SDEs and show how jets can lead to intuitive representations of Itô SDEs, including three different types of drawings. We explain that the mainstream choice of Fisk-StratonovichMcShane calculus for stochastic differential geom...

Given a continuous Gaussian process x which gives rise to p-geometric rough path for p∈(2,3), and general y controlled by x, under proper conditions we establish the relationship between Skorohod integral ∫0tysd♢xs Stratonovich ∫0tysdxs. Our strategy is employ tools from paths theory Malliavin calculus analyze discrete sums of integrals.

We propose a numerical method for solving stochastic differential equations with dichotomous Markov noise. The numerical scheme is formulated such that (i) the stochastic formula used follows the Stratonovich-Taylor form over the entire range of noise correlation times, including the Gaussian white noise limit; and (ii) the method is readily applicable to dynamical systems driven by arbitrary t...